Sizes of InfinitiesDate: 01/31/97 at 06:22:45 From: Mark Clackum Subject: Formula for proving that 1 infinity can be larger than another What is the formula for proving that one infinity can be larger than another? The number of fractions between 1 and 1000 should be larger than the number of fractions between 1 and 2. Date: 01/31/97 at 08:52:48 From: Doctor Steven Subject: Re: Formula for proving that 1 infinity can be larger than another There are lots of ways to prove one infinity is "larger" than another infinity. For instance, consider the infinite sequence (ak): (ak) = 1,2,3,4,5,6,7,8.... Now consider the infinite sequence (bk): (bk) = 2,4,6,8,10.... Obviously, (ak) is all natural numbers, and (bk) is all even natural numbers. They both have an infinite number of terms, but (bk) is also a subset of (ak)! In fact (bk) looks to have 1/2 the number of terms as (ak). This is called the paradox of infinity and it comes from trying to use infinity as a number instead of as an idea. Comparing these sequences you would get 2*(infinity) = infinity. And in fact if you kept going you could eventually prove c*(infinity) = infinity, where c is any number. Looking at 2D points compared to 1D points you can get (infinity)^2 = infinity. Keep going in this way and you can get (infinity)^n = infinity. There do exist different types of infinity. This distinction is sort of hard to think about until it's explained. There are enumerable infinities and then there are non-countable infinities. The sequence of natural numbers is an enumerable infinity; if we sat down for an infinite length of time we could count from 1 to infinity. The number of points between 0 and 1 on the number line, however, is uncountably infinite. There is no way given an infinite length of time we could count the points. Where would we start? Pick a point and I can find another one closer to 0 than the one you picked. We say that uncountable infinite items are much larger than countable infinite items. The number of points between 0 and 1 on then number line is much greater than the size of the set of the natural numbers. I hope this helps; infinity is a hard subject to nail down. Here's a quote you might enjoy: "Infinity is like a stuffed walrus I can hold in the palm of my hand. Don't do anything with infinity you wouldn't do with a stuffed walrus." -Dr. Fletcher Va. Polytechnic Inst. and St. Univ. (i.e., don't use it as a number) -Doctor Steven, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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